A Simple Method to Reconstruct Firing Rates from Dendritic Calcium Signals

Calcium imaging using fluorescent reporters is the most widely used optical approach to investigate activity in intact neuronal circuits with single-cell resolution. Calcium signals, however, are often difficult to interpret, especially if the desired output quantity is membrane voltage or instantaneous firing rates. Combining dendritic intracellular electrophysiology and multi-photon calcium imaging in vivo, we recently investigated the relationship between optical signals recorded with the fluorescent calcium indicator Oregon Green BAPTA-1 (OGB-1) and spike output in principal neurons in the locust antennal lobe. We derived from these experiments a simple, empirical and easily adaptable method requiring minimal calibration to reconstruct firing rates from calcium signals with good accuracy and 50-ms temporal resolution.

Despite the wide availability of optical voltage reporters to estimate activity in intact neuronal circuits with single-cell resolution (Chanda et al., 2005;Cohen et al., 1978;Ferezou et al., 2006;Nuriya et al., 2006;Obaid et al., 1999;Sjulson and Miesenbock, 2006;Zhou et al., 2007), calcium imaging using fl uorescent reporters (Denk and Svoboda, 1997;Helmchen and Waters, 2002;Smetters et al., 1999;Tsien, 1999;Yuste and Katz, 1991) remains the most used optical approach to achieve similar goals. The robustness of exogenous and endogenous labeling with calcium reporters and the minimal impact they have on neuronal viability make them, in principle, excellent tools to probe activity at the level of neuron populations, single neurons, or single sub-cellular compartments (e.g., dendrites, spines, presynaptic boutons). The availability of genetically engineered calcium indicators (Mank and Griesbeck, 2008;Miyawaki et al., 1997Miyawaki et al., , 1999Reiff et al., 2005) and the possibility of controlling their selective expression (Feng et al., 2000;Hebert and McConnell, 2000;Jefferis et al., 2001Jefferis et al., , 2007Zong et al., 2005) reinforce their use and promise. Despite important remaining issues related to indicator response kinetics (see below), the constant refi nement of fl uorescent calcium indicator proteins (Garaschuk et al., 2007;Mank et al., 2006;Mao et al., 2008;Wallace et al. 2008), combined with improvements in optical microscopy techniques (Deisseroth et al., 2006;Dombeck et al., 2007;Engelbrecht et al., 2008;Holekamp et al., 2008;Llewellyn et al., 2008) makes calcium imaging a major tool to elucidate activity and function in neuronal circuits (Carlsson et al. 2005;Friedrich and Korsching, 1997;Galizia et al., 1999;Holekamp et al., 2008;Ohki et al., 2005;Sobel and Tank, 1994;Spors et al., 2006;Tabor et al., 2008;Wachowiak and Cohen, 2001;Wang et al. 2003Wang et al. , 2004. Often deemphasized, however, is the fact that calcium signals are not always straightforward to interpret, especially if the desired output quantity is membrane voltage or instantaneous fi ring rates. This study (and its source, Moreaux and Laurent, 2007) proposes a simple empirical method to estimate instantaneous fi ring rates of single neurons from fl uores-

PROBLEMS ARISING WHEN CONVERTING CALCIUM SIGNALS INTO VOLTAGE OR FIRING RATES
In the study of neuronal systems, signals from calcium indicators are, given an appropriately selected indicator with adequate calcium sensitivity, usually interpreted as indicating spiking activity; whereas calcium signals and voltage are generally indeed correlated, calcium signals can also be generated by sub-threshold voltage events or infl ux through ionotropic ion channels (Goldberg et al., 1999;Kovalchuk et al., 2000;Lin et al., 2007;Perez-Reyes, 2003;Soler-Llavina and Sabatini, 2006). Conversely, an indicator's sensitivity may not be matched to the fi ring statistics of the neurons under study. Correctly interpreting a calcium signal -being in a position to separate synaptic inputs from spike output for example, or estimating the fraction of spike output detected -thus depends on a judicious match between the indicator's sensitivity and the properties of the neurons under study. Often, this match is diffi cult or impossible to optimize, for lack of information on the neurons sampled, or for lack of prior testing, especially in the precise conditions in which imaging will be used (the operating range of neurons can differ dramatically between in vitro and in vivo conditions, for example). This match becomes increasingly important as the desired temporal resolution of activity estimates approaches the temporal scale of single action potentials. We note that neuronal activity often consists simply of modulations around baselines rates (themselves sometimes elevated, e.g., 30 spikes/s, or as low as 0) to instantaneous peaks up to several hundred spikes/s. In short, one should be both careful and realistic about what can be reasonably achieved.
To make possible a reasonable extraction of spiking activity, the relationship between calcium signals and voltage should be carefully addressed in a neuron-specifi c manner. We believe that this is fundamental for a correct functional interpretation of imaging data. Plainly, the functional calibration of indicators should be done in the cell type in which calcium will later be used as a substitute for electrical recordings, and ideally, in the same physiological conditions. Such conditions are harsh, probably even sometimes impossible. Yet, if we do not keep in mind their necessity, we run the risk of having to later re-interpret calcium-imaging data, causing unnecessary revisions.
Recently, the correspondence between odorevoked, somatic and dendritic calcium signals and spike output has been evaluated in vivo in Drosophila projection neurons (PNs) expressing the genetically encoded calcium reporter G-CaMP 1.3 (Jayaraman and Laurent, 2007). A recent investigation combining calcium imaging and voltage recordings in a nose and brain preparation of Xenopus laevis revealed the underestimated cell-specifi city of the correlation between spontaneous and odor-evoked spiking activity and mitral and granule cells' somatic calcium signals (Lin et al., 2007). In vivo somatic calcium signals were combined with simultaneous voltage recordings in anesthetized rats to identify temporally sparse spontaneous activity (∼1 Hz) and compared to somatic calcium recordings alone in awake animals (Greenberg et al., 2008). Spiking activity was restricted to single action potentials or short bursts; an original template-matching method (Greenberg et al., 2008, SI) applied to the calcium signal alone successfully extracted spike times with good accuracy. In adult zebrafi sh brain and nose explants, in a study combining soma calcium-signal and voltage recordings, Yaksi and Friedrich (2006) used a deconvolution method to extract spiking activity of sensory neurons stimulated with odors. The method assumed that each action potential gives rise to a stereotyped somatic calcium transient with mono-exponential decay over time, and that calcium summation caused by a train of action potentials is described well by a simple convolution of a mono-exponential kernel with the spike-train. This approach gave good fi ring-rate reconstructions with image sampling rates of ∼30 Hz on neuronal populations of adult zebrafi sh (Tabor et al., 2008;Yaksi and Friedrich, 2006), a remarkable achievement.
Note that most past attempts to extract spiking activity from single neurons have been performed using the fl uorescence of calcium indicators recorded at the soma. In invertebrates such as insects, spike-related calcium/fl uorescence transients are usually smaller from the soma (sometimes even absent, as in locust PNs; Moreaux and Laurent, 2007) than from dendrites; importantly, they can also be slower (∼1 s vs. ∼50 ms, decay times), making the capture of neuronal fi ring modulation questionable. Usually neurons' somata are spread out in space (more so in larger brains) hindering the capture of a complete neuronal population, even with fast scanning imaging techniques Holekamp et al., 2008;Reddy et al., 2008), and imposing non-ideal trade-offs between spatio-temporal resolution and signal/noise. Each approach should thus be tailored to its experimental system; all methods do not fi t all experimental and biological situations. Here we focus on one system, the locust olfactory system, and propose an easily modifi ed method to extract fi ring-rate estimates from calcium-dendritic signals recorded in vivo using the common calcium indicator OGB-1.

DENDRITIC CALCIUM IMAGING IN EARLY OLFACTORY CIRCUITS
The early olfactory system -olfactory bulb (vertebrates) or antennal lobe (insects) -with its well organized glomerular anatomy offers a beautiful opportunity to use dendritic calcium imaging to help decipher basic rules of sensory processing (Buck and Axel, 1991;Hallem et al., 2004;Mombaerts et al., 1996;Vosshall et al., 2000). In the exploration of such a sensory system, intrinsic and calcium imaging of glomeruli have been used in vivo to investigate spatio-temporal activity patterns of input, output and local neurons. Typically, imaging of glomeruli using various kinds of fl uorescent calcium indicators has been used as a proxy for electro-physiological recordings (Carlsson et al., 2005;Fried et al., 2002;Friedrich and Korsching, 1997;Galizia et al., 1999;Sachse and Galizia, 2002;Spors et al., 2006;Wachowiak and Cohen, 2001;Wang et al., 2003). An important point to elucidate in insect antennal lobe circuits is the true relationship between spike output and imaging data in PNs, the antennal lobe's sole output. We address this issue in locust, combining dendritic voltage recordings and two-photon calcium imaging in vivo during odor stimulation in intact, nonanesthetized animals (Figure 1). Our goal was to identify the main causes of calcium signal fl uctuations, to determine whether calcium signals might be used to estimate PNs spike output, and if so, with what accuracy.
Locust PNs are multiglomerular neurons, with projections to 10 or so individual glomeruli (∼30 μm diameter each). The adult locust antennal lobe contains about 1,000 glomeruli, distributed in a sphere of about 300 μm diameter. The glomeruli visited by individual PNs, however, always lie in a plane. By correctly orienting the preparation, all the glomeruli of a stained PN can thus be observed in one plane of focus. The planes defi ned by all PNs are parallel to one another. For our experiments, a fi ne borosilicate micropipette was back-fi lled with OGB-1, gently advanced into the neuropil and used to impale one PN, usually in its principal omega-shaped dendrite, or in a secondary dendrite close to a glomerular tuft ( Figure 1A). Recordings could often be held for several tens of minutes. Odors were presented to the animal in 1 s long pulses, and 24 s of calcium-indicator and intracellular-voltage data were collected (500 Hz image sampling) around the odor stimulus pulse (early fragment shown in Figures 1B,C). Our data thus contained long epochs of baseline activity, stimulus-evoked responses and recovery periods. PN electrical activity recorded in these experiments was identical (baseline activity; response characteristics; peak fi ring rates; range of response types; oscillatory activity) to that previously described, using different recording methods (soma intracellular recordings: Laurent and Davidowitz, 1994;Laurent et al., 1996;tetrode extracellular recordings: Mazor and Laurent, 2005;Perez-Orive et al., 2002).

THE MULTIPLE COMPONENTS OF THE GLOMERULAR DENDRITIC CALCIUM SIGNAL
Locust PN dendritic calcium signals are composed of multiple, distinct components. Our experiments suggested that at least three factors (Figure 2A) contribute to the variation of calcium in a glomerular dendrite (Moreaux and Laurent, 2007): calcium transients triggered by action potentials, voltage-dependent calcium entry caused by sub-threshold depolarization, and calcium clearance. First, we observed that a single action potential always produced a detectable calcium increase; single-spike calcium decay time was fast (∼50 ms) after pharmacological blockade of nicotinic-Ach-receptor mediated synaptic input, but slower and non-stereotyped (variable across the glomerular dendrites of a single PN for example) in intact conditions. Spike trains always led to a calcium signal summation. Second, spontaneous or stimulus-evoked sub-threshold depolarizations caused, from a membrane voltage of about −60 mV, a detectable calcium signal ( Figure 2B). While insect nicotinic receptors are known to be calcium permeable, much of this sub-threshold calcium signal seemed to be voltage-dependent, for it could be reproduced simply by depolarizing-current injection. By contrast with observations from somatic imaging, calcium clearance after summation in the dendrite was not necessarily mono-exponential. This might be due to local hyper-polarizing inputs from inhibitory pathways. Because these multiple contributions to the calcium signal sampled may interact in complicated (or at least poorly understood) ways, the extraction of a PN spike output is clearly not a straightforward exercise, especially if attaining single-spike resolution is required. We now summarize an empirical method for instantaneous fi ring-rate estimation, based on the above data and knowledge. Two-photon imaging optical technique allowing imaging of endogenous or exogenous fl uorescent molecules in cellular compartments in vivo in a non-invasive manner.

Moreaux and Laurent
From calcium to fi ring rate

A SIMPLE METHOD TO ESTIMATE SPIKE OUTPUT FROM DENDRITIC CALCIUM SIGNALS
Having access to simultaneously recorded voltage and fl uorescence, we investigated (Moreaux and Laurent, 2007) whether a PN's fi ring rate could be extracted from its glomerular calcium signal recorded at high sampling rate (500 Hz) using empirically derived routines selective for spikedependent signals. The goal was to transform the fl uorescence of the calcium indicator into spiking activity, using consistent rules and ultimately, a small number of experiment-specifi c measure-ments that require no physiological recording. We managed to estimate a PN's fi ring activity with a 50-ms time resolution from a single fl uorescence recording, using one experiment-specifi c parameter and a minimal neuron-specifi c calibration. In its principle, our estimation method relies on two assumptions: (1) when calcium summation occurs, the intracellular calcium value is linearly related to the average fi ring rate of the spike-train causing calcium integration, and (2) calcium decays are due either to cessation of fi ring or to inhibition. With this in mind, the main steps of Moreaux and Laurent From calcium to fi ring rate our algorithm are the following (see fl owchart in Figure 3). A fl uorescence single trial is smoothed with a Gaussian fi lter, to "enhance" the fl uorescence plateaus (or peaks) and the end of the fl uorescence-decays (called valleys). The level of fl uorescence representing the absence of fi ring at baseline (F B , see Figure 4) needs to be determined, so that the assumed linear relationship between the optical signal and fi ring can be used. F B is the rest value of the spike-independent components, due mainly to net, depolarizing synaptic bombardment. The fl uorescence trace is then normalized to this value. Then, peaks and valleys are detected to defi ne segments of activity.
Practically, one has to check that the fi rst peak of the recording comes after the fi rst valley and the last peak comes after the last valley. The value read for a given valley is then subtracted from all following fl uorescence values, until the next peak is reached. This step sets all the ends of decay to 0 and guarantees a coherent proportionality between the normalized fl uorescence and the actual fi ring rate while not yet accounting for calcium summation. The values from a peak to the next valley are reset to 0 using a simple Gaussian-decay function. Calcium summation is now taken into consideration by comparing the time interval between a valley and the previous peak to a characteristic time, T c , the duration over which spike-evoked calcium summation is not relevant. If this time interval <T c , summation must be factored in: the valley value is added to the following values until the next peak. Only positive values of the reconstruction are kept and scaling is applied. Setting a threshold around the average spontaneous spiking activity fi lters out large time-resolved synaptic events.

TWO CALIBRATION PARAMETERS (T C AND S)
A mapping between calcium signals and voltage (or fi ring rates) appropriate for integrative studies is probably attainable, regardless of the methodology, only if some prior calibration steps have been carried out in the same preparation and experimental conditions. In practice, because simultaneous voltage and calcium recordings in vivo in non-anesthetized animals tend to be diffi cult, the calibration steps should be as limited as possible. Our method relies on the knowledge of three parameters. The fi rst, F B (see Figure 4), is experiment-specifi c and requires no prior calibration (i.e., no paired imaging and electrophysiology). The other two (time parameter, T c , and scaling factor, S), are the only two parameters requiring calibration through prior paired imaging and electrophysiology. With locust PNs and the calcium indicator OGB-1, these two parameters could be estimated satisfactorily from paired simultaneous voltage and calcium recordings in fi ve different animals and by optimizing the overall fi t between predicted (from calcium signal) and measured (with a microelectrode) fi ring rates (Figure 4). Once the calibration done, our method was applied to predict PN fi ring patterns from calcium signals, using different paired recordings as tests. We obtained ∼80-90% mean accuracy. In each experiment, F B was the only experiment-specifi c parameter required (Figure 5).

SOURCES OF VARIABILITY
When exogenous calcium indicators are used, a key issue for fi ring-rate-estimate accuracy is the consistency of cytoplasmic concentration of the indicator across experiments. With bolus loading techniques, great care (usually acquired after much practice) is required to minimize inter-experiment   variations. With multi-cellular labeling, spatial gradients of calcium indicator concentrations may clearly create problems . The time and scaling parameters are related to the calcium-buffering capacity added to the neuron's own by the indicator. In our hands, following strict experimental procedures, the acrossexperiment variance of the time parameter T c was very small (∼5%). The major source of error originated from the variance of the scaling parameter, due to a deviation from the assumed linearity between calcium elevation and fi ring rate. There are at least two distinct potential sources of nonlinearity: (1) from the fl uorescent calcium indicator itself and (2) from intrinsic cellular properties. The latter especially makes defi ning a ceiling an empirical issue. With locust PNs and OGB1, we observed that the deviation from linear prediction was usually small for short-lived bursts of high-fi ring rate (PNs rarely fi re at rates >60 sp/s), but larger (and always and underestimate) during prolonged periods of fi ring, presumably due to a contribution of sustained synaptic inputs and sub-linear summation with spike-related calcium entry. Under pharmacological blockade of synaptic input (in conditions admittedly without much use for network issues), the variance of the scaling factor S was consistent with that of the time parameter measured in normal in vivo condition. Non-linear summation of spike-related events at high-fi ring rates and the superposition of sub-threshold and spike-dependent components, presumably non-linear, are probably responsible for the variance of the scaling factor.

CHOOSING THE RIGHT METHOD
If activity is sparse and stereotyped (e.g., short burst of few action potentials), if calcium decay time is shorter than the minimum inter-spike (or burst) interval, and if the correlation between calcium elevation and spiking has been properly established in a cell-specifi c manner (see Lin et al., 2007 for a warning), then template-matching methods applied to somatic calcium signals can provide a good estimate of the "on/off " state of a single or population of neurons (Greenberg et al. 2008). In the presence of sustained spiking activity, as in sensory receptor neurons for example, the task of estimating fi ring rates from calcium signals can be more delicate. The deconvolution approach of Yaksi and Friedrich (2006) assumes that spike-related calcium-transients are stereotyped. Deconvolution is performed in the frequency domain, as a high-pass fi lter applied to the de-noised calcium spectrum. A calibration step requires the determination of the average fi lter parameters from simultaneous voltage and calcium recordings. Like all inverse problems, this method is very sensitive to signal-to-(highfrequency) noise ratio. It is also computationally intensive. Finally, the required stereotypy of calcium decay is probably met mainly when fast calcium fl uctuations are damped, as expected in somata. Our approach is the opposite. No assumption is made about calcium decays associated with interruptions of fi ring. We impose a simple reset of those segments, detected in the time domain, and take calcium summation into account following a very simple rule. This method is computationally trivial and less sensitive to noise: we detect extrema rather than invert matrices in the frequency domain. Reconstruction accuracy is ∼80-90% on average -i.e., comparable to Yaksi and Friedrich's method (2006) -using raw calcium signals recorded at dendritic sites. The simplicity of the approach should be an advantage if large datasets are to be analyzed. The counterpoint is that our method so far rests on acquiring fast/dendritic calcium signals. Those are potentially well suited for systems such as the olfactory bulb and antennal lobe, where processing occurs in compact glomeruli. How the method fares in different systems, where dendrites are more widespread, remains to be tested After the specifi c PN/OGB-1 calibration, the algorithm may be applied to single PN-odor calcium imaging alone, baseline F B being the only trialspecifi c parameter, as illustrated here. Left: raw fl uorescence traces recorded from one PN in response to six different odors (stimulus during gray bar). Stipple line indicates F B for each trial. Right: predicted (from left, green) and measured (cyan) smoothed fi ring rates. An overall 80-90% mean accuracy is obtained.